Problem: What is the least positive integer that has a remainder of 0 when divided by 2, a remainder of 1 when divided by 3, and a remainder of 2 when divided by 4?
Let $a$ be the desired number. We know that \begin{align*}
a & \equiv 0\pmod 2\\
a & \equiv 1\pmod 3\\
a & \equiv 2\pmod 4
\end{align*} Note that $a \equiv 2\pmod 4$ automatically implies $a \equiv 0\pmod 2$, so only $a \equiv 1\pmod 3$ and $a \equiv 2\pmod 4$ need to be considered. The first few positive solutions of $a \equiv 2\pmod 4$ are $2,6,10$. While the first two do not satisfy $a \equiv 1\pmod 3$, luckily $\boxed{10}$ does!